# Likelihood ratio test for GENLINMIXED in SPSS?

I am analyzing some linguistic data using SPSS 25, and have encountered some problems. Please note that I am not a statistician, so excuse me if my question is trivial. Basically, I am using the Mixed Models/Generalized Linear option to analyze my data. The model converges, but the output does not contain -2LL statistics (only AIC and BIC) based on which I could do a Likelihood Ratio test , and compare my models using a chi-squared table. SPSS does produce this statistic though, if I use the Linear Mixed Models option, Instead of the GLMM, but I specifically need GLMMs, as my response variable is ordinal. I also do not quite understand what kind of p-values GENLINMIXED reports in the "Fixed Effects" window. They are consistent with my alternative hypothesis (what I believed to be a significant predictor of my response is significant, and vice versa), and it would be great if I could report them, but as far as I know, in the case of mixed models, one can only determine whether an effect is significant by using the Likelihood Ratio test and compare the model with the effect in question to a null model without the effect. Can anyone shed some light on what exactly SPSS is doing in this case, and why?

A couple of notes:

• Most often the p-values given next to regression coefficients are based on the Wald test statistic, which is the estimated value of the coefficient divided by its standard error. This tests the null hypothesis that the specific coefficient is zero, and all other coefficients are non-zero.
• You could test the same hypothesis also with a likelihood ratio test. Asymptotically (i.e., for large enough sample size) the two tests are equivalent giving you the same p-value. However, in smaller samples there are differences that most often boil down in the distribution used to derive the p-value; for example, for normal outcome data and small sample sizes, the t distribution is more appropriate than the normal distribution.
• As far as I know, GLMMs in SPSS are fitted using the penalized quasi likelihood method, which is known to be sub-optimal and produce biased results, especially for binary data or count data with low expected counts. A better alternative is to use the adaptive Gaussian quadrature that is available in other software.
• Thanks for taking your time to answer, Dimitris! I can now see that the Wald statistic and the Likelihood Ratio are closely related, and if I understand correctly, it should be okay to report the Wald p-values, as the Wald statistic only relies on the unrestricted model, so there is no need for comparing nested models. What I still don't understand is why SPSS does not report -2LL along with AIC and BIC in the GLMM output, while it reports it for regular linear mixed models. My hunch is that it is not possible to calculate -2LL using the quasi-likelihood method, is this correct? Thanks! – Bálint L. Tóth Jun 2 '19 at 10:58